Fractals and self-similarity

Iterated function systems
Mandelbrot sets I guess
Non-local derivatives
Long memory time series
In nature
relationship to dimensions
Estimating fractal dimension
References

From the original patent application of the original “fractal” vise

Objects with a fractional Hausdorff dimension, AFAICT. For some equivocation on this theme, see wikipedia.

Iterated function systems

Connection to design grammars.

Fun fact: there is a fractal renderer built into GIMP.

Mandelbrot sets I guess

TBD

Non-local derivatives

See fractional derivatives.

Long memory time series

See, for example, fractional brownian motion, long memory time series.

In nature

Pattern formation in nature often looks fractal to some approximation, or at some range of scales.

Horse lung

Starfish of infinite surface area.

relationship to dimensions

David S. Richeson, A Mathematician’s Guided Tour Through High Dimensions does a nice job of relating fractals to measure-theoretic notions of dimension.

Estimating fractal dimension

Question: How closely related is this to estimating a Hurst exponent? How close to grammatical induction? Various classic methods based on naïve plug-in versions of mathematical definitions are given in Theiler (1990). A new one, which I am curious about, is Chamorro-Posada (2016) based on some kind of compression argument, basically, gzipping copies of the image that have been downsampled by various ratios and watching how the file size changes as a kind of entropy estimate.

I am reminded of Cosma Shalizi’s cautionary note on estimating Entropies and Information using Lempel-Ziv, and the caveat:

Jose M. Amigo, Janusz Szczepanski, Elek Wajnryb and Maria V. Sanchez-Vives, “Estimating the Entropy Rate of Spike Trains via Lempel-Ziv Complexity”, Neural Computation 16 (2004): 717--736: Normally, I have strong views on using Lempel-Ziv to measure entropy rates, but here they are using the 1976 Lempel-Ziv definitions, not the 1978 ones. The difference is subtle, but important; 1978 leads to gzip and practical compression algorithms, but very bad entropy estimates; 1976 leads, as they show numerically, to reasonable entropy rate estimates, at least for some processes. Thanks to Dr. Szczepanski for correspondence about this paper.]

References

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